The proof of Lemma  is straightforward, and is provided in the first part of the proof of Proposition 1 in [1]}. Together, Lemma  and Lemma  enable the comparison of the subgraph density matrix \(\mathbf {S}\)  and the mean subgraph density matrix \(\mathbf {M}\)  and moreover shed light on the relationship between the principal components output by the PCAN and sPCAN methods. These relationships are made clear in Proposition  and Theorem .
